3.31.47 \(\int \frac {x^4 (-q+p x^4) \sqrt {q+p x^4}}{b x^8+a (q+p x^4)^4} \, dx\)

Optimal. Leaf size=457 \[ -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {x \left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt {p x^4+q}}{-\sqrt [4]{a} p x^4-\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^4+q}}{\sqrt [4]{a} p x^4+\sqrt [4]{a} q-\sqrt [4]{b} x^2}\right )}{8 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{a} p x^4}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{a} q}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{x \sqrt {p x^4+q}}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{a} p x^4}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{a} q}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{x \sqrt {p x^4+q}}\right )}{8 a^{3/8} b^{5/8}} \]

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Rubi [F]  time = 2.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^8 + a*(q + p*x^4)^4),x]

[Out]

q*Defer[Int][(x^4*Sqrt[q + p*x^4])/(-(b*x^8) - a*(q + p*x^4)^4), x] + p*Defer[Int][(x^8*Sqrt[q + p*x^4])/(b*x^
8 + a*(q + p*x^4)^4), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx &=\int \left (\frac {q x^4 \sqrt {q+p x^4}}{-a q^4-4 a p q^3 x^4-b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8-4 a p^3 q x^{12}-a p^4 x^{16}}+\frac {p x^8 \sqrt {q+p x^4}}{a q^4+4 a p q^3 x^4+b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8+4 a p^3 q x^{12}+a p^4 x^{16}}\right ) \, dx\\ &=p \int \frac {x^8 \sqrt {q+p x^4}}{a q^4+4 a p q^3 x^4+b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8+4 a p^3 q x^{12}+a p^4 x^{16}} \, dx+q \int \frac {x^4 \sqrt {q+p x^4}}{-a q^4-4 a p q^3 x^4-b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8-4 a p^3 q x^{12}-a p^4 x^{16}} \, dx\\ &=p \int \frac {x^8 \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx+q \int \frac {x^4 \sqrt {q+p x^4}}{-b x^8-a \left (q+p x^4\right )^4} \, dx\\ \end {align*}

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Mathematica [C]  time = 7.80, size = 15065, normalized size = 32.96 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^8 + a*(q + p*x^4)^4),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 24.94, size = 425, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^4}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^4}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{a} q}{\sqrt [8]{b}}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b} x^2}{\sqrt [8]{a}}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{a} p x^4}{\sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{a} q}{\sqrt [8]{b}}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b} x^2}{\sqrt [8]{a}}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{a} p x^4}{\sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^8 + a*(q + p*x^4)^4),x]

[Out]

-1/8*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^4])/(a^(1/4)*q - b^(1/4)*x^2
+ a^(1/4)*p*x^4)])/(a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q +
p*x^4])/(a^(1/4)*q - b^(1/4)*x^2 + a^(1/4)*p*x^4)])/(8*a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTanh[((Sqrt[1
- 1/Sqrt[2]]*a^(1/8)*q)/b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*b^(1/8)*x^2)/a^(1/8) + (Sqrt[1 - 1/Sqrt[2]]*a^(1/8)*p*
x^4)/b^(1/8))/(x*Sqrt[q + p*x^4])])/(8*a^(3/8)*b^(5/8)) - (Sqrt[2 + Sqrt[2]]*ArcTanh[((Sqrt[1 + 1/Sqrt[2]]*a^(
1/8)*q)/b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*b^(1/8)*x^2)/a^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*a^(1/8)*p*x^4)/b^(1/8))/(x*
Sqrt[q + p*x^4])])/(8*a^(3/8)*b^(5/8))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.21, size = 47, normalized size = 0.10

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (16 \textit {\_Z}^{8} a +b \right )}{\sum }\frac {\ln \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right ) \sqrt {2}}{64 a}\)	\(47\)
elliptic	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (16 \textit {\_Z}^{8} a +b \right )}{\sum }\frac {\ln \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right ) \sqrt {2}}{64 a}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x,method=_RETURNVERBOSE)

[Out]

1/64/a*sum(1/_R^5*ln(1/2*(p*x^4+q)^(1/2)*2^(1/2)/x-_R),_R=RootOf(16*_Z^8*a+b))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )} x^{4}}{b x^{8} + {\left (p x^{4} + q\right )}^{4} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x, algorithm="maxima")

[Out]

integrate(sqrt(p*x^4 + q)*(p*x^4 - q)*x^4/(b*x^8 + (p*x^4 + q)^4*a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^4\,\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{a\,{\left (p\,x^4+q\right )}^4+b\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*(q + p*x^4)^(1/2)*(q - p*x^4))/(a*(q + p*x^4)^4 + b*x^8),x)

[Out]

-int((x^4*(q + p*x^4)^(1/2)*(q - p*x^4))/(a*(q + p*x^4)^4 + b*x^8), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(p*x**4-q)*(p*x**4+q)**(1/2)/(b*x**8+a*(p*x**4+q)**4),x)

[Out]

Timed out

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